1.5 State Transformation of the LTI system（特征值）andEigenvector1.5. 1 Eigenvalue（特征向量）Consider the LTI system described bythe state equationX = AX + BuWhere XeR"is the state vector, AeRnxnis the systemmatrixandplays animportant rolein system properties.When u = 0 , the system given by (1.99) is calledafreesystem（自由系统）(1.99)X = AX当X和X方向相同但大小不同的时候，引入标量因子几，得到AX = ^X(-A)X =0有非零解iff-A=0（I一A）为A的特征矩阵（characteristicmatrix)
1.5 State Transformation of the LTI system 1.5.1 Eigenvalue （特征值）and Eigenvector （特征向量） Consider the LTI system described by the state equation X = AX + Bu Where is the state vector, is the system matrix and plays an important role in system properties. When , the system given by (1.99) is called a free system（自由系统） n X R nn AR u = 0 X = AX (1.99) AX = X 当 X 和 X 方向相同但大小不同的时候，引入标量因子 ，得到 (I − A)X = 0 有非零解iff I A − = 0 ( ) I A − 为A的特征矩阵（characteristic matrix）
DefinitionsThe polynomial about △ is called the characteristic polynomial（特征多项式）0(2) =|ul - A| = " +Zα,2(1.101)i=0Q(a)=aI -A|=0 is called the characteristic equation （特征方程)If the polynomial Q(a) can be written in factored form as(1.102)Q(a) = det(al - A) = IT(a - a,)i=1The roots 2, (i = l,2,..:,n) of the characteristic equationarecalledtheeigenvalues（特征值/特征根）ofA
The polynomial about is called the characteristic polynomial （特征多项式） − = = − = + 1 0 ( ) n i i i n Q I A (1.101) Q I A ( ) 0 = − = is called the characteristic equation（特征方程） If the polynomial can be written in factored form as Q() = = − = − n i Q i 1 () det(I A) ( ) (1.102) The roots of the characteristic equation are called the eigenvalues （特征值/特征根）of A (i 1,2, ,n) i = Definitions
Propertiesabouteigenvalue1. If the elements of A are real, then its eigenvalues are eitherrealorincomplexconjugatepairs.2. If 2, (i = l,2,.".,n) are the eigenvalues of A, thentr(A)=≥,(1.103)i-l3. If , (i=1,2,...,n) are the eigenvalues of A, then they are theeigenvalues of AT .4. If A is nonsingular, with eigenvalues a, (i = l,2,..,n) , then,1aretheeigenvalues of A-l元
Properties about eigenvalue 1. If the elements of A are real, then its eigenvalues are either real or in complex conjugate pairs. 2. If are the eigenvalues of A, then (i 1,2, ,n) i = = = n i i 1 tr(A) (1.103) 4. If A is nonsingular, with eigenvalues , then, (i 1,2, ,n) i = 1 i are the eigenvalues of 1 A − 3. If are the eigenvalues of A, then they are the eigenvalues of . (i 1,2, ,n) i = T A
Any nonzero vector V which satisfies the matrix equation(1.104)(2,I - A)V, = 0iscalledtheeigenvector（特征向量）ofAassociatedwitheigenvalues,(i =1,2,.,n)注：特征向量常用于状态变换（statetransformation）的计算。自学：广义特征向量的计算如果A只含有单根，可直接由Eq.(1.104)求出对应的特征向量如果A有重根且非对称，则由Eq.(1.105)求出所有的广义特征向量(AI-A)V1 = 0(a, I - A)Vi2 = -Vi1( I - A)Vi3 = -Vi2(1.105).(21 - A)Vim = -V(m-)
Any nonzero vector Vi which satisfies the matrix equation (i I − A)Vi = 0 (1.104) is called the eigenvector (特征向量) of A associated with eigenvalues (i 1,2, ,n) i = 注：特征向量常用于状态变换（ state transformation ）的计算。 如果 A只含有单根，可直接由Eq.(1.104)求出对应的特征向量. 如果 A有重根且非对称，则由Eq.(1.105)求出所有的广义特征向量. 自学：广义特征向量的计算 1 11 1 12 11 1 13 12 1 1 1( 1) ( ) 0 ( ) ( ) ( ) m m I A V I A V V I A V V I A V V − − = − = − − = − − = − (1.105)
1.5.2StateTransformation（状态变换）Consider theLTl system described by (1.107)X(t) = AX(t) + Bu(t)(1.107)y(t) = CX(t) + Du(t)注：由于状态X(t)的选取不唯一（not unique），那么它所对应的状态空间描述也不唯一。linearstatetransformationX→XDefineanXnnonsingular（非奇异）matrixP，andletX(t) = PX(t)(1.108)X(t) = P-IX(t)(1.109)Eq.(1.108) or Eq.(1.109) is called the linear nonsingular statetransformation（线性非奇异状态变换）：
1.5.2 State Transformation（状态变换） Consider the LTI system described by (1.107) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + (1.107) 注：由于状态X(t)的选取不唯一（not unique），那么它所对应的状态空间 描述也不唯一。 X X → linear state transformation Define a n×n nonsingular （非奇异） matrix P, and let X(t) = PX(t) ( ) ( ) 1 X t P X t − = (1.108) (1.109) Eq.(1.108) or Eq.(1.109) is called the linear nonsingular state transformation （线性非奇异状态变换）
By substituting(1.108)and(1.109)intoLTl system(1.107),wehavethenew stateequationintermsof thenewstateX(1.110)X(t) = P-' APX(t) + P- Bu(t) = AX(t) + Bu(t)A = P-' AP → similarity transformation of A(1.111)where(1.112)B = P-'BBy substituting (1.108) into (1.107), we have the output equation interms of the new state Xy(t) = CPX(t) + Du(t) = CX(t) + Du(t)(1.113)C =CP(1.114)where(1.115)D=D进而得到系统关于状态 X 的状态空间描述 (A,B,C,D)应用：对于给定的状态空间描述A,B,C,D}，我们可以通过一个线性状态变换 X(t)=PX(t)来求出变换后的新的状态空间描述(A,B,C,D)特别的，可以是约当规范形或对角规范形
By substituting (1.108) and (1.109) into LTI system (1.107), we have the new state equation in terms of the new state X ( ) ( ) ( ) ( ) ( ) 1 1 X t = P APX t + P Bu t = AX t + Bu t − − (1.110) where A P AP −1 = similarity transformation of A (1.111) B P B −1 = (1.112) By substituting (1.108) into (1.107), we have the output equation in terms of the new state X y(t) = CPX(t) + Du(t) = CX(t) + Du(t) (1.113) where C = CP (1.114) D = D (1.115) 进而得到系统关于状态 X 的状态空间描述 { , , , } A B C D 应用：对于给定的状态空间描述{A,B,C,D}，我们可以通过一个线性状 态变换 来求出变换后的新的状态空间描述 ，特 别的，可以是约当规范形或对角规范形。 X(t) = PX(t) { , , , } A B C D
1.5.3softheStateInvariancePropertiesTransformation（状态变换的不变性）The characteristic equation, eigenvalues, eigenvectors andtransfer function are all preserved by the nonsingularstate transformation.特征方程、特征值、特征向量和传递函数在状态变换下是保持不变的
1.5.3 Invariance Properties of the State Transformation （状态变换的不变性） •The characteristic equation, eigenvalues, eigenvectors and transfer function are all preserved by the nonsingular state transformation. 特征方程、特征值、特征向量和传递函数在状态变换下 是保持不变的
X(t) = PX(t)X(t) = AX(t) + Bu(t)X(t) = P-'APX(t) + P-'Bu(t) = AX(t) + Bu(t)y(t) = CX(t) + Du(t)y(t)= CPX(t) + Du(t) = CX(t)+ Du(t)1、 The characteristic equation of the system (A,B,C,D)sI-A=sI-P-"AP=sP-"P-P-"AP=P-'(sI-A)P= 0sI - A = |P-I|sI - A|P| = sI - A| = 0注：特征方程在状态变换下是保持不变的，因此特征值、特征向量在状态变换下也是保持不变的
1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t t − − = + = + = + = + X P APX P Bu AX Bu ( ) ( ) ( ) y CPX Du CX Du ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + X(t) = PX(t) 1、The characteristic equation of the system { , , , } A B C D ( ) 0 1 1 1 1 − = − = − = − = − − − − sI A sI P A P sP P P A P P sI A P 0 1 − = − = − = − sI A P sI A P sI A 注：特征方程在状态变换下是保持不变的，因此特征值、特征向量 在状态变换下也是保持不变的
2、 The transfer function of the system described by(A, B,C, D)G(s)=C(sI - A)-'B +D =CP(sI - P-"AP)-'P-B+ D= C[P(sI - P-'AP)P-'}-"B + D =C[P(sP-'P- P-'AP)P-I}-'B +D=C[PP-'(sI - A)PP-'I-'B + D = C(sI - A)-'B + D= G(s)注：传递函数在状态变换下是保持不变的。事实上，能控性（controllability）和能观性（observability）在状态变换下是保持不变的
2、The transfer function of the system described by { , , , } A B C D ( ) [ ( ) ] ( ) [ ( ) ] [ ( ) ] ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s sI s s s s s s G C P P A P P B D C I A B D C P I P A P P B D C P P P P A P P B D G C I A B D C P I P A P P B D = = − + = − + = − + = − + = − + = − + − − − − − − − − − − − − − − − 注：传递函数在状态变换下是保持不变的。 事实上，能控性（controllability）和能观性（observability）在状态 变换下是保持不变的
X(t) = AX(t) + Bu(t)y(t) = CX(t) + Du(t)X(t) = PX(t)Nonsingular lineartransformationX(t) = AX(t) + Bu(t) = P-'AP X(t)+P-'B u(t)y(t)=CX(t)+ Du(t) =CP X(t)+D u(t)Invariance Properties: The characteristic equation, eigenvalues,eigenvectors andtransferfunction
( ) ( ) ( ) ( ) ( ) ( ) t t t t t t = + = + X AX Bu y CX Du ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + X(t) = PX(t) Nonsingular linear transformation 1 1 ( ) ( ) t t − − = + P AP X P B u = + CP X D u ( ) ( ) t t Invariance Properties: The characteristic equation, eigenvalues, eigenvectors and transfer function